When physicists write expressions involving spinors $\psi \in S \otimes V$, where $S=S_+ \oplus S_-$ is a complex spinor representation of a spin group $Spin(2d)$ and $V$ is a complex representation of a non-Abelian Lie group $G$ preserving an inner product, they often continue to make use of explicit bases for the Clifford algebra. What is the standard mathematically invariant way of writing and defining expressions like $\bar \psi, \psi^\dagger, \psi^*, \psi^\dagger \gamma^0, \psi^\dagger \gamma^0 \gamma^5, etc... $ without using such explicit bases, but using only

The (presumably Hermitian) inner product on $S$.

The (presumably Hermitian) inner product on $V$.

Tensor products and direct sums of vector spaces and their elements.

Invariants inside the Clifford Algebra and other algebraic structures etc...

In particular, what is the standard mathematically invariant definition of the spinorial source term $J(\psi)$ in the Yang-Mills Equation
$$
d_A^* F_A = J(\psi)
$$ where $V$ represents a non-trivial representation of a non-abelian $G$?